commit 4dcf6389aefd295070c3a96934287983f6929387 parent 41939c270b1e003da69bb4cbf6976c0d696272da Author: miksa234 <milutin@popovic.xyz> Date: Wed, 8 Dec 2021 16:14:44 +0100 done hw04 Diffstat:
37 files changed, 669 insertions(+), 1434 deletions(-)
diff --git a/sesh4/src/.ipynb_checkpoints/Popovic_assingment4-checkpoint.ipynb b/sesh4/src/.ipynb_checkpoints/Popovic_assingment4-checkpoint.ipynb @@ -1,352 +0,0 @@ -{ - "cells": [ - { - "cell_type": "code", - "execution_count": 67, - "id": "a0245b45", - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "rank_7_CPD" - ] - }, - "execution_count": 67, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "using LinearAlgebra\n", - "using Plots\n", - "using Distributions\n", - "using LaTeXStrings\n", - "using Random\n", - "using TSVD\n", - "include(\"../../sesh3/src/functions.jl\")\n", - "include(\"../../sesh2/src/functions.jl\")" - ] - }, - { - "cell_type": "markdown", - "id": "4c5e0a42", - "metadata": {}, - "source": [ - "# Exercise 1" - ] - }, - { - "cell_type": "code", - "execution_count": 135, - "id": "f8ec20e3", - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "tucker_eval (generic function with 1 method)" - ] - }, - "execution_count": 135, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "function hosvd(C, r)\n", - " S0 = copy(C)\n", - " d = length(size(C))\n", - " n = size(C)\n", - " V = []\n", - " singular_values = []\n", - " errors = []\n", - " for k=1:d\n", - " B = reshape(S0, (prod([[r_i for r_i in r[1:(k-1)]]..., [i_k for i_k in n[(k+1):end]]...])..., n[k]))\n", - " U_hat, Sig_hat, V_hat = tsvd(convert(Matrix{Float64}, B), r[k])\n", - " B_hat = U_hat * Diagonal(Sig_hat) * transpose(V_hat)\n", - " W_hat = B * V_hat\n", - " S0 = reshape(W_hat , ([r_i for r_i in r[1:k]]..., [n_k for n_k in n[k+1:end]]...))\n", - " \n", - " append!(V, [V_hat])\n", - " append!(singular_values, [Sig_hat])\n", - " C_hat = tucker_eval(S0, V)\n", - " append!(errors, norm(C_hat - C)/norm(C))\n", - " end\n", - " return V, S0, singular_values, errors\n", - "end\n", - "\n", - "\n", - "function tucker_eval(S, V)\n", - " d = length(V)+1\n", - " A = copy(S)\n", - " for (k, V_k) in enumerate(reverse(V))\n", - " A = mode_k_dot(A, V_k, d-k)\n", - " end\n", - " return A\n", - "end" - ] - }, - { - "cell_type": "code", - "execution_count": 136, - "id": "7794ccdd", - "metadata": {}, - "outputs": [], - "source": [ - "d = 4;\n", - "n = [20+k for k=1:d];\n", - "r = [2*k for k=1:d];\n", - "V = [rand(Uniform(-1, 1), n[k], r[k]) for k=1:d];\n", - "S = rand(Uniform(-1, 1), r...);\n", - "A = tucker_eval(S, V);\n", - "V, S, singular_values, errors = hosvd(A, r);" - ] - }, - { - "cell_type": "markdown", - "id": "c0f93f64", - "metadata": {}, - "source": [ - "# Exercise 2 " - ] - }, - { - "cell_type": "code", - "execution_count": 70, - "id": "0562e4e7", - "metadata": {}, - "outputs": [], - "source": [ - "function f(x)\n", - " s = 0\n", - " for (k, xk) in enumerate(x)\n", - " s = s .+ (xk.^2 / (8^(k-1)))\n", - " end\n", - " return (1 .+ s).^(-1)\n", - "end\n", - "\n", - "function g(x)\n", - " s1 = 0\n", - " s2 = 0\n", - " for (k, xk) in enumerate(x)\n", - " s1 = s1 .+ (xk.^2 / (8^(k-1)))\n", - " s2 = s2 .+ (4 * pi * xk)/(4^(k-1))\n", - " end\n", - " return sqrt.(s1) .* (1 .+ 1/2 * cos.(s2))\n", - "end\n", - "\n", - "function unfold(A, k)\n", - " B = permutedims(A, vcat(k, setdiff(1:length(size(A)), k)))\n", - " d = size(B, 1)\n", - " return reshape(B, (d, div(length(B),d)))\n", - "end\n", - "\n", - "t(n) = [(2*(i-1)/(n-1) - 1) for i in 1:n]\n", - "\n" - ] - }, - { - "cell_type": "markdown", - "id": "13abb65b", - "metadata": {}, - "source": [ - "# Exercise 3" - ] - }, - { - "cell_type": "code", - "execution_count": 80, - "id": "c08e0c7f", - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "rank_approx (generic function with 2 methods)" - ] - }, - "execution_count": 80, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "eps_j = [1/(10^(j*2)) for j=1:6]\n", - "function rank_approx(C, eps_j=eps_j)\n", - " r_jk = []\n", - " singular_vec = []\n", - " d = length(size(C))\n", - " eps_jk = []\n", - " for k=1:d\n", - " C_k = tenmat(C, k)\n", - " U, S, V = svd(C_k)\n", - " append!(singular_vec, [S])\n", - " for (j, e_j) in enumerate(eps_j)\n", - " for r=1:d\n", - " U_hat, S_hat, V_hat = tsvd(C_k, r)\n", - " C_k_hat = U_hat * Diagonal(S_hat) * transpose(V_hat)\n", - " if norm(C_k_hat-C_k)/norm(C_k) <= e_j\n", - " append!(eps_jk, norm(C_k_hat-C_k)/norm(C_k))\n", - " append!(r_jk, r)\n", - " break\n", - " end\n", - " end\n", - " end\n", - " end\n", - " return reshape(r_jk, (length(eps_j), d)), reshape(eps_jk, (length(eps_j), d)), singular_vec\n", - "end" - ] - }, - { - "cell_type": "code", - "execution_count": 103, - "id": "21f29099", - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "false" - ] - }, - "execution_count": 103, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "d = 4\n", - "A = [f([t_i, t_j, t_k, t_l]) for t_i in t(d), t_j in t(d), t_k in t(d), t_l in t(d)];\n", - "B = [g([t_i, t_j, t_k, t_l]) for t_i in t(d), t_j in t(d), t_k in t(d), t_l in t(d)];" - ] - }, - { - "cell_type": "markdown", - "id": "8f1a72d4", - "metadata": {}, - "source": [ - "# Exercise 4, 5" - ] - }, - { - "cell_type": "code", - "execution_count": 371, - "id": "65e2e226", - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "tt_svd (generic function with 1 method)" - ] - }, - "execution_count": 371, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "function ttmps_eval(U, n)\n", - " A = U[1]\n", - " for U_k in U[2:end]\n", - " A = A * U_k\n", - " end\n", - " return reshape(A, n)\n", - "end\n", - "\n", - "function tt_svd(A, n, r, d)\n", - " r_0 = 1\n", - " r_new = [r_0, r...]\n", - " S_0_hat = copy(A)\n", - " C = []; singular_val = []; errors = []\n", - " for k=2:d\n", - " B_k = reshape(S_0_hat, (r_new[k-1] * n[k-1], prod([n[i] for i=k:d])))\n", - " U_hat, Sig_hat, V_hat = tsvd(convert(Matrix{Float64}, B_k), r_new[k])\n", - " C_k = reshape(U_hat, (r_new[k-1], n[k-1], r_new[k]))\n", - " W_k_hat = Diagonal(Sig_hat) * transpose(V_hat)\n", - " S_0_hat = reshape(W_k_hat, (r_new[k], [n[i] for i=k:d]...))\n", - " append!(C, [C_k])\n", - " append!(singular_val, [Sig_hat])\n", - " \n", - " #A_hat = ttmps_eval(S_0_hat, C)\n", - " #append!(errors, norm(A_hat - A)/norm(A))\n", - " end\n", - " append!(C, [S_0_hat])\n", - " return C, singular_val, errors\n", - "end" - ] - }, - { - "cell_type": "markdown", - "id": "bad26f36", - "metadata": {}, - "source": [ - "# Exercise 6" - ] - }, - { - "cell_type": "code", - "execution_count": 372, - "id": "d86060bd", - "metadata": {}, - "outputs": [], - "source": [ - "d = 4;\n", - "n = [20+k for k=1:d];\n", - "r = [[2*k for k=1:(d-1)]..., 1];\n", - "V = [rand(Uniform(-1, 1), n[k], r[k]) for k=1:d];\n", - "S = rand(Uniform(-1, 1), r...);\n", - "A = tucker_eval(S, V);\n", - "C, singular_val, errors = tt_svd(A, n, r, d);\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "3d398f32", - "metadata": {}, - "outputs": [], - "source": [] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "d3f2138c", - "metadata": {}, - "outputs": [], - "source": [] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "e39ada8d", - "metadata": {}, - "outputs": [], - "source": [] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "9f06c365", - "metadata": {}, - "outputs": [], - "source": [] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Julia 1.6.3", - "language": "julia", - "name": "julia-1.6" - }, - "language_info": { - "file_extension": ".jl", - "mimetype": "application/julia", - "name": "julia", - "version": "1.7.0" - } - }, - "nbformat": 4, - "nbformat_minor": 5 -} diff --git a/sesh4/src/Popovic_assingment4.ipynb b/sesh4/src/Popovic_assingment4.ipynb @@ -1,1003 +0,0 @@ -{ - "cells": [ - { - "cell_type": "code", - "execution_count": 281, - "id": "a0245b45", - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "rank_7_CPD" - ] - }, - "execution_count": 281, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "using LinearAlgebra\n", - "using Plots\n", - "using Distributions\n", - "using LaTeXStrings\n", - "using Random\n", - "using TSVD\n", - "using TensorOperations\n", - "include(\"../../sesh3/src/functions.jl\")\n", - "include(\"../../sesh2/src/functions.jl\")" - ] - }, - { - "cell_type": "markdown", - "id": "3aa3c01c", - "metadata": {}, - "source": [ - "# Popovic Milutin: Solution to Assingment 4" - ] - }, - { - "cell_type": "markdown", - "id": "4c5e0a42", - "metadata": {}, - "source": [ - "# Exercise 1" - ] - }, - { - "cell_type": "code", - "execution_count": 321, - "id": "f8ec20e3", - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "tucker_eval (generic function with 1 method)" - ] - }, - "execution_count": 321, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "function hosvd(C, r)\n", - " S0 = copy(C)\n", - " d = length(size(C))\n", - " n = size(C)\n", - " V = []\n", - " singular_values = []\n", - " errors = []\n", - " for k=1:d\n", - " B = reshape(S0, (prod([[r_i for r_i in r[1:(k-1)]]..., [i_k for i_k in n[(k+1):end]]...])..., n[k]))\n", - " U_hat, Sig_hat, V_hat = tsvd(convert(Matrix{Float64}, B), r[k])\n", - " B_hat = U_hat * Diagonal(Sig_hat) * transpose(V_hat)\n", - " W_hat = B * V_hat\n", - " S0 = reshape(W_hat , ([r_i for r_i in r[1:k]]..., [n_k for n_k in n[k+1:end]]...))\n", - " \n", - " append!(V, [V_hat])\n", - " append!(singular_values, [Sig_hat])\n", - " C_hat = tucker_eval(S0, V)\n", - " append!(errors, norm(C_hat - C)/norm(C))\n", - " end\n", - " return V, S0, singular_values, errors\n", - "end\n", - "\n", - "\n", - "function tucker_eval(S, V)\n", - " d = length(V)+1\n", - " A = copy(S)\n", - " for (k, V_k) in enumerate(reverse(V))\n", - " A = mode_k_dot(A, V_k, d-k)\n", - " end\n", - " return A\n", - "end" - ] - }, - { - "cell_type": "markdown", - "id": "5bcc3aaf", - "metadata": {}, - "source": [ - "# Exercise 2" - ] - }, - { - "cell_type": "code", - "execution_count": 121, - "id": "7794ccdd", - "metadata": {}, - "outputs": [], - "source": [ - "d = 4;\n", - "n = [20+k for k=1:d];\n", - "r = [2*k for k=1:d];\n", - "V = [rand(Uniform(-1, 1), n[k], r[k]) for k=1:d];\n", - "S = rand(Uniform(-1, 1), r...);\n", - "A = tucker_eval(S, V);\n", - "V, S, singular_values, errors = hosvd(A, r);" - ] - }, - { - "cell_type": "markdown", - "id": "c0f93f64", - "metadata": {}, - "source": [ - "# Exercise 3" - ] - }, - { - "cell_type": "code", - "execution_count": 336, - "id": "0562e4e7", - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "t (generic function with 1 method)" - ] - }, - "execution_count": 336, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "function f(x)\n", - " s = 0\n", - " for (k, xk) in enumerate(x)\n", - " s = s .+ (xk.^2 / (8^(k-1)))\n", - " end\n", - " return (1 .+ s).^(-1)\n", - "end\n", - "\n", - "function g(x)\n", - " s1 = 0\n", - " s2 = 0\n", - " for (k, xk) in enumerate(x)\n", - " s1 = s1 .+ (xk.^2 / (8^(k-1)))\n", - " s2 = s2 .+ (4 * pi * xk)/(4^(k-1))\n", - " end\n", - " return sqrt.(s1) .* (1 .+ 1/2 * cos.(s2))\n", - "end\n", - "\n", - "function unfold(A, k)\n", - " B = permutedims(A, vcat(k, setdiff(1:length(size(A)), k)))\n", - " d = size(B, 1)\n", - " return reshape(B, (d, div(length(B),d)))\n", - "end\n", - "\n", - "function sing_unfold(C, d)\n", - " vals = []\n", - " for k=1:d\n", - " U, S, V = svd(unfold(C, k))\n", - " append!(vals, [S])\n", - " end\n", - " return vals\n", - "end\n", - "\n", - "t(n) = [(2*(i-1)/(n-1) - 1) for i in 1:n]" - ] - }, - { - "cell_type": "code", - "execution_count": 656, - "id": "9b066c20", - "metadata": {}, - "outputs": [], - "source": [ - "d = 4; n = 51;\n", - "A = [f([t1, t2, t3, t4]) for t1 in t(n), t2 in t(n), t3 in t(n), t4 in t(n)];\n", - "B = [g([t1, t2, t3, t4]) for t1 in t(n), t2 in t(n), t3 in t(n), t4 in t(n)];" - ] - }, - { - "cell_type": "markdown", - "id": "dec133ac", - 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xlabel=L\"n\",\n", - " lw = 2,\n", - " yaxis=:log,\n", - " title=\"Unfolding of A\")\n", - " plot!(p[2],\n", - " sing_b[k],\n", - " ylabel=L\"$\\log(\\Sigma_{B_k})$\",\n", - " label=L\"$\\Sigma_{_B%$k}$\",\n", - " xlabel=L\"n\",\n", - " lw = 2,\n", - " yaxis=:log,\n", - " title=\"Unfolding of B\")\n", - "end\n", - "plot(p)" - ] - }, - { - "cell_type": "code", - "execution_count": 657, - "id": "378a53b9", - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "(Any[6 4 3 2; 10 9 6 3; … ; 22 22 17 12; 25 28 24 16], Any[0.0013630857628283304 0.009518329263607794 0.0014509541080803049 0.0023517835675429264; 7.159528543404029e-5 5.3592722281017756e-5 3.786849235801887e-5 6.886783883231526e-5; … ; 2.3048992932560663e-9 4.865446890113614e-9 9.055790097998212e-9 4.494606310730708e-9; 7.057122859935684e-11 4.462051964776192e-11 4.58120456032028e-11 9.80285039910442e-11])" - ] - }, - "execution_count": 657, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "eps_j = [1/(10^(j*2)) for j=1:5]\n", - "function rank_approx(C, eps_j=eps_j)\n", - " d = length(size(C))\n", - " eps_jk = []; r_jk = []; singular_vec = []\n", - " for k=1:d\n", - " C_k = unfold(C, k)\n", - " for (j, e_j) in enumerate(eps_j)\n", - " for r=1:rank(C_k)\n", - " U_hat, S_hat, V_hat = tsvd(C_k, r)\n", - " C_k_hat = U_hat * Diagonal(S_hat) * transpose(V_hat)\n", - " if norm(C_k_hat-C_k)/norm(C_k) <= e_j\n", - " append!(eps_jk, norm(C_k_hat-C_k)/norm(C_k))\n", - " append!(r_jk, r)\n", - " break\n", - " end\n", - " end\n", - " end\n", - " end\n", - " return reshape(r_jk, (length(eps_j), d)), reshape(eps_jk, (length(eps_j), d))\n", - "end\n", - "r_jk_a, eps_jk_a = rank_approx(A)\n", - "r_jk_b, eps_jk_b = rank_approx(B)\n" - ] - }, - { - "cell_type": "code", - "execution_count": 658, - "id": "c5c7dddd", - "metadata": {}, - "outputs": [ - { - "data": { - "image/svg+xml": [ - "<?xml version=\"1.0\" encoding=\"utf-8\"?>\n", - "<svg xmlns=\"http://www.w3.org/2000/svg\" 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"markdown", - "id": "900d297f", - "metadata": {}, - "source": [ - "# Exercise 3b" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "21f29099", - "metadata": {}, - "outputs": [], - "source": [] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "2cf1c0d3", - "metadata": {}, - "outputs": [], - "source": [] - }, - { - "cell_type": "markdown", - "id": "8f1a72d4", - "metadata": {}, - "source": [ - "# Exercise 4, 5" - ] - }, - { - "cell_type": "code", - "execution_count": 305, - "id": "65e2e226", - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "tt_svd (generic function with 1 method)" - ] - }, - "execution_count": 305, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "function ttmps_eval(U, n)\n", - " A = U[1]\n", - " for U_k=U[2:end]\n", - " A = contract(A, U_k)\n", - " end\n", - " return reshape(A, n...)\n", - "end\n", - "\n", - "function tt_svd(A, n, r, d)\n", - " r_0 = 1\n", - " r_new = [r_0, r...]\n", - " S_0_hat = copy(A)\n", - " S0s = []\n", - " C = []; singular_val = []; errors = []\n", - " for k=2:d\n", - " B_k = reshape(S_0_hat, (r_new[k-1] * n[k-1], prod([n[i] for i=k:d])))\n", - " U_hat, Sig_hat, V_hat = tsvd(convert(Matrix{Float64}, B_k), r_new[k])\n", - " C_k = reshape(U_hat, (r_new[k-1], n[k-1], r_new[k]))\n", - " W_k_hat = Diagonal(Sig_hat) * transpose(V_hat)\n", - " S_0_hat = reshape(W_k_hat, (r_new[k], [n[i] for i=k:d]...))\n", - " \n", - " append!(C, [C_k])\n", - " append!(singular_val, [Sig_hat])\n", - " A_hat = ttmps_eval([C..., S_0_hat], n )\n", - " append!(errors, norm(A_hat - A)/norm(A))\n", - " end\n", - " append!(C, [S_0_hat])\n", - " return C, singular_val, errors\n", - "end" - ] - }, - { - "cell_type": "markdown", - "id": "bad26f36", - "metadata": {}, - "source": [ - "# Exercise 6" - ] - }, - { - "cell_type": "code", - "execution_count": 306, - "id": "d86060bd", - "metadata": {}, - "outputs": [], - "source": [ - "d = 4;\n", - "n = [20+k for k=1:d];\n", - "r = [[2*k for k=1:(d-1)]..., 1];\n", - "V = [rand(Uniform(-1, 1), n[k], r[k]) for k=1:d];\n", - "S = rand(Uniform(-1, 1), r...);\n", - "A = tucker_eval(S, V);\n", - "C, singular_val, errors= tt_svd(A, n, r, d);" - ] - }, - { - "cell_type": "markdown", - "id": "b444a67d", - "metadata": {}, - "source": [ - "# Exercise 7" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "2533bc95", - "metadata": {}, - "outputs": [], - "source": [] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "d3f2138c", - "metadata": {}, - "outputs": [], - "source": [] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "e39ada8d", - "metadata": {}, - "outputs": [], - "source": [] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "9f06c365", - "metadata": {}, - "outputs": [], - "source": [] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "ec3859f2", - "metadata": {}, - "outputs": [], - "source": [] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "d71e44b5", - "metadata": {}, - "outputs": [], - "source": [] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Julia 1.6.3", - "language": "julia", - "name": "julia-1.6" - }, - "language_info": { - "file_extension": ".jl", - "mimetype": "application/julia", - "name": "julia", - "version": "1.7.0" - } - }, - "nbformat": 4, - "nbformat_minor": 5 -} diff --git a/sesh4/src/functions.jl b/sesh4/src/functions.jl @@ -0,0 +1,21 @@ +#/usr/bin/julia + +t(n) = [(2*(i-1)/(n-1) - 1) for i in 1:n] + +function f(x) + s = 0 + for (k, xk) in enumerate(x) + s = s .+ (xk.^2 / (8^(k-1))) + end + return (1 .+ s).^(-1) +end + +function g(x) + s1 = 0 + s2 = 0 + for (k, xk) in enumerate(x) + s1 = s1 .+ (xk.^2 / (8^(k-1))) + s2 = s2 .+ (4 * pi * xk)/(4^(k-1)) + end + return sqrt.(s1) .* (1 .+ 1/2 * cos.(s2)) +end diff --git a/sesh4/src/main.jl b/sesh4/src/main.jl @@ -0,0 +1,308 @@ +#/usr/bin/julia + +using LinearAlgebra +using Plots +using Distributions +using LaTeXStrings +using Random +using TSVD: tsvd # todo: implement +using TensorToolbox: tenmat, ttm, contract # todo:implement +using Plots.Measures + +include("mps-tt.jl") +include("tucker-hosvd.jl") +include("unf-aprx.jl") +include("functions.jl") + +function main() + # Exercise 2 + d = 4; + n = [20+k for k=1:d]; + r = [2*k for k=1:d]; + V = [rand(Uniform(-1, 1), n[k], r[k]) for k=1:d]; + S = rand(Uniform(-1, 1), r...); + C = tucker_eval(S, V); + Vs, S0, vals, errors = my_hosvd(C, r); + + p = plot(errors, + ylabel="error", + label=L"$\frac{\|\hat{A}_k - A\|_F}{\|A\|_F}$", + xlabel=L"k", + lw = 1, + xticks=collect(1:d), + markershape=:circle, + legend=:topleft, + margin=5mm, + dpi=300, + title="HOSVD of Uniform Tensor") + savefig(p, "./plots/hosvd-uniform-error.png") + + # Exercise 3 + d = 4; n = 51; + A = [f([t1, t2, t3, t4]) for t1 in t(n), t2 in t(n), t3 in t(n), t4 in t(n)]; + B = [g([t1, t2, t3, t4]) for t1 in t(n), t2 in t(n), t3 in t(n), t4 in t(n)]; + + # Exercise 3a + σ_unf_a = σ_unfold(A, d); + σ_unf_b = σ_unfold(B, d); + p1 = plot(margin=5mm) + p2 = plot(margin=5mm) + for k=1:d + plot!(p1, + σ_unf_a[k], + ylabel=L"$\log(\Sigma_{A_k})$", + label=L"$\Sigma_{A_%$k}$", + xlabel=L"n", + lw = 1, + yaxis=:log, + markershape=:circle, + dpi=300, + title="Unfolding of A") + plot!(p2, + σ_unf_b[k], + ylabel=L"$\log(\Sigma_{B_k})$", + label=L"$\Sigma_{_B%$k}$", + xlabel=L"n", + lw = 1, + dpi=300, + markershape=:circle, + yaxis=:log, + title="Unfolding of B") + end + savefig(p1, "./plots/singular-dec-A.png") + savefig(p2, "./plots/singular-dec-B.png") + + ϵ_s = [1/(10^(j*2)) for j=1:5] # computer not goode enough for j = 6 + + r_jk_a, ϵ_jk_a, σ_jk_a = rank_approx(A); + r_jk_b, ϵ_jk_b, σ_jk_b = rank_approx(B); + p1 = plot(size=[700, 350], margin=5mm, dpi=300) + p2 = plot(size=[700, 350], margin=5mm, dpi=300) + for k=1:d + plot!(p1, + 1 ./ ϵ_s, + r_jk_a[:, k], + ylabel=L"$r_{jk}$", + label=L"A-$\varepsilon_{j%$k}$", + xlabel=L"\varepsilon_{j}^{-1}", + lw = 2, + markershape=:circle, + xaxis=:log10, + legend=:topleft, + title=L"Tensor $f(x_1, x_2, x_3, x_4)$") + plot!(p2, + 1 ./ ϵ_s, + r_jk_b[:, k], + ylabel=L"$r_{jk}$", + label=L"B-$\varepsilon_{j%$k}$", + xlabel=L"\varepsilon_{j}^{-1}", + lw = 2, + markershape=:circle, + xaxis=:log10, + legend=:topleft, + title=L"Tensor $g(x_1, x_2, x_3, x_4)$") + end + savefig(p1, "./plots/hosvd-error-A.png") + savefig(p2, "./plots/hosvd-error-B.png") + + # Exercise 3b + Nj_hosvd_a = []; ϵ_hosvd_a = []; σ_hosvd_a = []; + Nj_hosvd_b = []; ϵ_hosvd_b = []; σ_hosvd_b = []; + + println("Checking validity of error analysis for HOSVD") + for j=1:length(ϵ_s) + S0_a, Vs_a, vals_a, errors_a = my_hosvd(A, r_jk_a[j, :]) + N_vals = 0 + for V in Vs_a + N_vals += length(V) + end + append!(Nj_hosvd_a, N_vals + length(S0_a)) + append!(ϵ_hosvd_a, errors_a[end]) + append!(σ_hosvd_a, [vals_a]) + println("ϵ_hosvd_a[j] <= |ϵ_jk_a|_f : $(ϵ_hosvd_a[j] <= norm(ϵ_jk_a)) ϵ_hosvd_a[j] = $(ϵ_hosvd_a[j])") + + S0_b, Vs_b, vals_b, errors_b = my_hosvd(B, r_jk_b[j, :]) + N_vals = 0 + for V in Vs_b + N_vals += length(V) + end + append!(Nj_hosvd_b, N_vals + length(S0_b)) + append!(ϵ_hosvd_b, errors_b[end]) + append!(σ_hosvd_b, [vals_b]) + println("ϵ_hosvd_b[j] <= |ϵ_jk_b|_f : $(ϵ_hosvd_b[j] <= norm(ϵ_jk_b)) ϵ_hosvd_b[j] = $(ϵ_hosvd_b[j])") + end + + # exercise 3b + j = 5 + p1 = plot(dpi=300, size=[800, 350], margin=5mm) + p2 = plot(dpi=300, size=[800, 350], margin=5mm) + for k=1:d + plot!(p1, σ_unf_a[k][1:r_jk_a[j, k]] ./ σ_hosvd_a[j][k], + ylabel=L"\frac{\sigma^{A_k}_\alpha}{\sigma^{hosvd}_{k\alpha}}", + label=L"k=%$k", + xlabel=L"j", + xticks=collect(1:length(r_jk_a)), + yticks=[1, 1.001], + ylim=[1, 1.001], + lw = 2, + markershape=:circle, + legend=:topleft, + title=L"$f(x_1, x_2, x_3, x_4)$") + plot!(p2, σ_unf_b[k][1:r_jk_b[j, k]] ./ σ_hosvd_b[j][k], + ylabel=L"\frac{\sigma^{B_k}_\alpha}{\sigma^{hosvd}_{k\alpha}}", + label=L"k=%$k", + xlabel=L"j", + yticks=[1, 1.001], + ylim=[1, 1.001], + lw = 2, + markershape=:circle, + legend=:topleft, + title=L"$g(x_1, x_2, x_3, x_4)$") + end + savefig(p1, "./plots/hosvd-sigmaratio-a.png") + savefig(p2, "./plots/hosvd-sigmaratio-b.png") + + # Exercise 3d + + p1 = plot(size=[700, 350], margin=5mm, dpi=300) + p2 = plot(size=[700, 350], margin=5mm, dpi=300) + plot!(p1, + 1 ./ ϵ_s, + Nj_hosvd_a, + ylabel=L"$N_{j}$", + label=L"A-$N_{j}$", + xlabel=L"\varepsilon_{j}^{-1}", + lw = 2, + markershape=:circle, + xaxis=:log10, + legend=:topleft, + title=L"Tucker of $f(x_1, x_2, x_3, x_4)$") + plot!(p2, + 1 ./ ϵ_s, + Nj_hosvd_b, + ylabel=L"$N_{j}$", + label=L"B-$N_{j}$", + xlabel=L"\varepsilon_{j}^{-1}", + lw = 2, + markershape=:circle, + xaxis=:log10, + legend=:topleft, + title=L"Tucker of $g(x_1, x_2, x_3, x_4)$") + savefig(p1, "./plots/hosvd-Nj-a.png") + savefig(p2, "./plots/hosvd-Nj-b.png") + + # exercise 6 + d = 4; + n = [20+k for k=1:d]; + r = [[2*k for k=1:(d-1)]..., 1]; + r_new = [1, r...] + V = [rand(Uniform(-1, 1), (r_new[i], n[i], r_new[i+1])) for i=1:d]; + D = ttmps_eval(V, n) + C, singular_val, errors= tt_svd(D, n, r, d); + p = plot(errors, + ylabel="error", + label=L"$\frac{\|\hat{A}_k - A\|_f}{\|A\|_f}$", + xlabel=L"k", + lw = 1, + xticks=collect(1:d), + markershape=:circle, + legend=:topleft, + margin=5mm, + dpi=300, + title="TT-SVD of uniform tensor") + savefig(p, "./plots/ttsvd-uniform-error.png") + + # exercise 7 + + d = 4; n = 51; + ndim = [n for i=1:d] + A = [f([t1, t2, t3, t4]) for t1 in t(n), t2 in t(n), t3 in t(n), t4 in t(n)]; + B = [g([t1, t2, t3, t4]) for t1 in t(n), t2 in t(n), t3 in t(n), t4 in t(n)]; + + # exercise 7b + Nj_ttsvd_a = []; ϵ_ttsvd_a = []; σ_ttsvd_a = []; + Nj_ttsvd_b = []; ϵ_ttsvd_b = []; σ_ttsvd_b = []; + + println("Checking validity of error analysis for TT-SVD") + for j=1:size(ϵ_jk_a, 1) + Vs_a, vals_a, errors_a = tt_svd(A, ndim, r_jk_a[j, :], d); + N_vals = 0 + for V in Vs_a + N_vals += length(V) + end + append!(Nj_ttsvd_a, N_vals) + append!(ϵ_ttsvd_a, errors_a[end]) + append!(σ_ttsvd_a, [vals_a]) + println("ϵ_ttsvd_a[j] <= |ϵ_jk_a|_f : $(ϵ_ttsvd_a[j] <= norm(ϵ_jk_a)) ϵ_ttsvd_a[j] = $(ϵ_ttsvd_a[j])") + + Vs_b, vals_b, errors_b = tt_svd(B, ndim, r_jk_b[j, :], d); + N_vals = 0 + for V in Vs_b + N_vals += length(V) + end + append!(Nj_ttsvd_b, N_vals) + append!(ϵ_ttsvd_b, errors_b[end]) + append!(σ_ttsvd_b, [vals_b]) + println("ϵ_ttsvd_b[j] <= |ϵ_jk_b|_f : $(ϵ_ttsvd_b[j] <= norm(ϵ_jk_b)) ϵ_ttsvd_b[j] = $(ϵ_ttsvd_b[j])") + end + + j = 5 + p1 = plot(dpi=300, size=[800, 350], margin=5mm) + p2 = plot(dpi=300, size=[800, 350], margin=5mm) + for k=1:d-1 + plot!(p1, σ_unf_a[k][1:r_jk_a[j, k]] ./ σ_ttsvd_a[j][k], + ylabel=L"\frac{\sigma^{A_k}_\alpha}{\sigma^{ttsvd}_{k\alpha}}", + label=L"k=%$k", + xlabel=L"j", + xticks=collect(1:length(r_jk_a)), + yticks=[1, 1000], + # ylim=[1, 1000], + lw = 2, + markershape=:circle, + legend=:topleft, + title=L"$f(x_1, x_2, x_3, x_4)$") + plot!(p2, σ_unf_b[k][1:r_jk_b[j, k]] ./ σ_ttsvd_b[j][k], + ylabel=L"\frac{\sigma^{B_k}_\alpha}{\sigma^{ttsvd}_{k\alpha}}", + label=L"k=%$k", + xlabel=L"j", + yticks=[1, 1000], + # ylim=[1, 1000], + lw = 2, + markershape=:circle, + legend=:topleft, + title=L"$g(x_1, x_2, x_3, x_4)$") + end + savefig(p1, "./plots/ttsvd-sigmaratio-a.png") + savefig(p2, "./plots/ttsvd-sigmaratio-b.png") + + # Exercise 7d + p1 = plot(size=[700, 350], margin=5mm, dpi=300) + p2 = plot(size=[700, 350], margin=5mm, dpi=300) + plot!(p1, + 1 ./ ϵ_s, + Nj_ttsvd_a, + ylabel=L"$N_{j}$", + label=L"A-$N_{j}$", + xlabel=L"\varepsilon_{j}^{-1}", + lw = 2, + markershape=:circle, + xaxis=:log10, + legend=:topleft, + title=L"MPS-TT of $f(x_1, x_2, x_3, x_4)$") + plot!(p2, + 1 ./ ϵ_s, + Nj_ttsvd_b, + ylabel=L"$N_{j}$", + label=L"B-$N_{j}$", + xlabel=L"\varepsilon_{j}^{-1}", + lw = 2, + markershape=:circle, + xaxis=:log10, + legend=:topleft, + title=L"MPS-TT of $g(x_1, x_2, x_3, x_4)$") + savefig(p1, "./plots/ttsvd-Nj-a.png") + savefig(p2, "./plots/ttsvd-Nj-b.png") + +end + +main() diff --git a/sesh4/src/mps-tt.jl b/sesh4/src/mps-tt.jl @@ -0,0 +1,40 @@ +#/usr/bin/julia + +using LinearAlgebra +using Plots +using Plots.Measures +using Distributions +using LaTeXStrings +using Random +using TSVD: tsvd # todo: implement +using TensorToolbox: tenmat, ttm, contract # todo:implement + +function ttmps_eval(U, n) + A = U[1] + for U_k=U[2:end] + A = contract(A, U_k) + end + return reshape(A, n...) +end + +function tt_svd(A, n, r, d) + r_0 = 1 + r_new = [r_0, r...] + S_0_hat = copy(A) + S0s = [] + C = []; singular_val = []; errors = [] + for k=2:d + B_k = reshape(S_0_hat, (r_new[k-1] * n[k-1], prod([n[i] for i=k:d]))) + U_hat, Sig_hat, V_hat = tsvd(convert(Matrix{Float64}, B_k), r_new[k]) + C_k = reshape(U_hat, (r_new[k-1], n[k-1], r_new[k])) + W_k_hat = Diagonal(Sig_hat) * transpose(V_hat) + S_0_hat = reshape(W_k_hat, (r_new[k], [n[i] for i=k:d]...)) + + append!(C, [C_k]) + append!(singular_val, [Sig_hat]) + A_hat = ttmps_eval([C..., S_0_hat], n) + append!(errors, norm(A_hat - A)/norm(A)) + end + append!(C, [S_0_hat]) + return C, singular_val, errors +end diff --git a/sesh4/src/plots/hosvd-Nj-a.png b/sesh4/src/plots/hosvd-Nj-a.png Binary files differ. diff --git a/sesh4/src/plots/hosvd-Nj-b.png b/sesh4/src/plots/hosvd-Nj-b.png Binary files differ. diff --git a/sesh4/src/plots/hosvd-error-A.png b/sesh4/src/plots/hosvd-error-A.png Binary files differ. diff --git a/sesh4/src/plots/hosvd-error-B.png b/sesh4/src/plots/hosvd-error-B.png Binary files differ. diff --git a/sesh4/src/plots/hosvd-sigmaratio-a.png b/sesh4/src/plots/hosvd-sigmaratio-a.png Binary files differ. diff --git a/sesh4/src/plots/hosvd-sigmaratio-b.png b/sesh4/src/plots/hosvd-sigmaratio-b.png Binary files differ. diff --git a/sesh4/src/plots/hosvd-uniform-error.png b/sesh4/src/plots/hosvd-uniform-error.png Binary files differ. diff --git a/sesh4/src/plots/singular-dec-A.png b/sesh4/src/plots/singular-dec-A.png Binary files differ. diff --git a/sesh4/src/plots/singular-dec-B.png b/sesh4/src/plots/singular-dec-B.png Binary files differ. diff --git a/sesh4/src/plots/ttsvd-Nj-a.png b/sesh4/src/plots/ttsvd-Nj-a.png Binary files differ. diff --git a/sesh4/src/plots/ttsvd-Nj-b.png b/sesh4/src/plots/ttsvd-Nj-b.png Binary files differ. diff --git a/sesh4/src/plots/ttsvd-sigmaratio-a.png b/sesh4/src/plots/ttsvd-sigmaratio-a.png Binary files differ. diff --git a/sesh4/src/plots/ttsvd-sigmaratio-b.png b/sesh4/src/plots/ttsvd-sigmaratio-b.png Binary files differ. diff --git a/sesh4/src/plots/ttsvd-uniform-error.png b/sesh4/src/plots/ttsvd-uniform-error.png Binary files differ. diff --git a/sesh4/src/tucker-hosvd.jl b/sesh4/src/tucker-hosvd.jl @@ -0,0 +1,45 @@ +#/usr/bin/julia + +using LinearAlgebra +using Plots +using Distributions +using LaTeXStrings +using Random +using TSVD: tsvd # todo: implement +using TensorToolbox: tenmat, ttm, contract # todo:implement +using Plots.Measures + +function my_hosvd(C, r) + S0 = copy(C); d = length(size(C)); n = size(C) + V = []; singular_values = []; errors = []; + for k=1:d + # reshape & permute S_{k-1} -> B + B_perm = permutedims(S0, ([[i for i=1:d if i!=k]..., k])) + B = reshape(B_perm, (prod([[r_i for r_i in r[1:(k-1)]]..., [i_k for i_k in n[(k+1):d]]...])..., n[k])) + B = convert(Matrix{Float64}, B) + + # rank r_k T-SVD of B -> B^hat + U_hat, Sig_hat, V_hat = tsvd(B, r[k]) + B_hat = U_hat * Diagonal(Sig_hat) * transpose(V_hat) + + # reshape & permute B^hat -> S_k + W_hat = B_hat * V_hat + W_reshape = reshape(W_hat , ([r_i for r_i in r[1:(k-1)]]..., [n_k for n_k in n[k+1:end]]..., r[k])) + S0 = permutedims(W_reshape , ([i for i=1:k-1]..., d, [i for i=k:d-1]...)) + + + append!(V, [V_hat]) + append!(singular_values, [Sig_hat]) + C_hat = tucker_eval(S0, V) + append!(errors, norm(C_hat - C)/norm(C)) + end + return V, S0, singular_values, errors +end + +function tucker_eval(S, V) + d = length(V); A = copy(S) + for (k, V_k) in enumerate(V) + A = ttm(A, V_k, k) + end + return A +end diff --git a/sesh4/src/unf-aprx.jl b/sesh4/src/unf-aprx.jl @@ -0,0 +1,42 @@ +#/usr/bin/julia + +using LinearAlgebra +using Plots +using Distributions +using LaTeXStrings +using Random +using TSVD: tsvd # todo: implement +using TensorToolbox: tenmat, ttm, contract # todo:implement +using Plots.Measures + +function σ_unfold(C, d) + Σ_s = [] + for k=1:d + U, Σ, V = svd(tenmat(C, k)) + append!(Σ_s, [Σ]) + end + return Σ_s +end + +ϵ_s = [1/(10^(j*2)) for j=1:5] # computer not goode enough for j = 6 +function rank_approx(C, ϵ_s=ϵ_s) + d = length(size(C)) + ϵ_jk = []; r_jk = []; σ_jk = [] + for k=1:d + C_k = tenmat(C, k) + for (j, ϵ_j) in enumerate(ϵ_s) + for r=1:rank(C_k) + U_hat, Σ_hat, V_hat = tsvd(C_k, r) + C_k_hat = U_hat * Diagonal(Σ_hat) * transpose(V_hat) + if norm(C_k_hat-C_k)/norm(C_k) <= ϵ_j + append!(ϵ_jk, norm(C_k_hat-C_k)/norm(C_k)) + append!(r_jk, r) + append!(σ_jk, [Σ_hat]) + break + end + end + end + end + ndims = (length(ϵ_s), d) + return reshape(r_jk, ndims), reshape(ϵ_jk, ndims), reshape(σ_jk, ndims) +end diff --git a/sesh4/tex/main.pdf b/sesh4/tex/main.pdf Binary files differ. diff --git a/sesh4/tex/main.tex b/sesh4/tex/main.tex @@ -22,9 +22,13 @@ \usepackage{float} \usepackage{yhmath} \usepackage{tikz} +\usepackage{algorithm} +\usepackage{algpseudocode} \usetikzlibrary{patterns,decorations.pathmorphing,positioning} \usetikzlibrary{calc,decorations.markings} +\newcommand{\eps}{\varepsilon} + \usepackage[backend=biber, sorting=none]{biblatex} \addbibresource{uni.bib} @@ -57,100 +61,230 @@ \begin{document} \maketitle \tableofcontents -\section{Assignment 3} -\subsection{Implementing the CP-ALS Algorithm} -The main idea of the algorithm is that we have a rank $R\in \mathbb{N}$ CPD -of a tensor $A \in \mathbb{R}^{n_1 \times \cdot n_d}$ given in terms of -$U_k \in \mathbb{R}^{n_k \times R}$, for $\mathbb{N}\ni k>2$, $k = -\{1,\dots,d\}$ and for $n_1, \dots, n_d\; \in \mathbb{N}$. We want to find a -rank $r\in \mathbb{N}$ CPD of the tensor $A$, by taking an initial guess -for some $V_k \in \mathbb{R}^{n_k \times r}$ and updating it for each $k$ by -optimizing the problem +\section{Assignment 4} +\subsection{HOSVD Algorithm} +The HOSVD algorithm is used to compute the Tucker approximation of a given tensor $A \in \mathbb{R}^{n_1 +\times \cdots \times n_d}$ with given ranks $r_1, \dots, r_d\ \in \mathbb{N}$ ($d +\in \mathbb{N}$ and $n_1, \dots, n_d\ \in \mathbb{N}$. Additionally we would +like to compute and save +\begin{itemize} + \item the Forbenius norms of the error produced at each step of the + algorithm $\frac{\| A - \hat{A}_k\|_F}{\|A\|_F}$ and the vectors of + \item the vector of singular values directly approximated by the + algorithm +\end{itemize} +The Tucker decomposition for $A$, with the ranks above is the following \begin{align} - \phi(V_1, \dots, V_d) \rightarrow \min_{V_k \in \mathbb{R}^{n_k\times - r}}, + A_{i_1,\cdots,i_d} = \sum_{\alpha_1=1}^{r_1} \cdot + \sum_{\alpha_d=1}^{r_d} (U_1)_{i_1,\alpha_1} \cdots (U_d)_{i_d,\alpha_d} + S_{\alpha_1, \dots, \alpha_d}, \end{align} -where $\phi(V_1, \dots, V_d)$ is the error function, determined by +where $U_k \in \mathbb{R}^{n_k \times r_k}$ for all $k \in \{1, \cdots, d\}$ +and $S \in \mathbb{R}^{r_1 \times \cdots \times r_d}$ is called the +Tucker-core. + +The HOSVD algorithm with the additional requirements for the Tucker +decomposition of $A$ is the following +\begin{algorithm}[H] + \caption{HOSVD algorithm}\label{alg: hosvd} +\begin{algorithmic} + \State $\hat{S}_0 \gets A$ + \State $A_0 \gets A$ + \For{$k = 1, \dots, d$} + \State $(B_k)_{\alpha_1\cdots\alpha_{k-1}\cdot i_{k+1} \cdots + i_d,i_k} \gets (\hat{S}_{k-1})_{\alpha_1, \dots, \alpha_{k-1}, i_k, + \dots, i_d}$ \Comment{permute then reshape} + + \State $\hat{B}_k \gets \hat{U}_k \hat{\Sigma}_k \hat{V}_k^*$ + \Comment{rank $r_k$ T-SVD for $B_k$} + + \State $(\hat{S}_k)_{\alpha_{1}, \dots, \alpha_{k}, i_{k+1},\dots,i_d + } \gets (B_k \hat{V}_k)_{\alpha_{1}, \dots, \alpha_{k-1}, i_{k+1}, \dots, + i_d, \alpha_k}$ \Comment{reshape then permute} + + \State $(A_k)_{i_1,\cdots,i_d} \gets \sum_{\alpha_1=1}^{r_1} \cdot + \sum_{\alpha_k=1}^{r_k} (\hat{V}_1)_{i_1,\alpha_1} \cdots (\hat{V}_k)_{i_d,\alpha_d} + S_{\alpha_1, \dots, \alpha_{k}, i_{k+1}, \dots, i_d}$ + + \State \text{save} $\frac{\| A - \hat{A}_k\|_F}{\|A\|_F}$ + \State \text{save} $\hat{V}_k$ + \State \text{save} $\hat{\Sigma}_k$ + \EndFor +\end{algorithmic} +\end{algorithm} +We note that at the $k-th$ step of the algorithm the shapes of the tensors +are +\begin{align} + \hat{S}_{k-1} &\in \mathbb{R}^{r_1 \times \cdots \times r_{k-1} \times n_k + \times \cdots n_d} \\ + \hat{V}_k &\in \mathbb{R}^{n_k \times r_k},\\ + \hat{B}_k &\in \mathbb{R}^{r_1\cdots r_{k-1}\cdot n_{k+1} \cdots n_d, + \times n_k},\\ + \hat{\Sigma}_k &\in \mathbb{R}^{r_k \times 1}. +\end{align} +\subsection{Testing the HOSVD} +For the case $d=4$ we construct a quasirandom Tucker decomposition by drawing +the entries for $U_k \in \mathbb{R}^{n_k \times r_k}$ and $S \in \mathbb{R}^{r_1 +\times \cdots \times r_d}$ uniformly on $[-1, 1]$. The output of the errors in +the $k-th$ steps is in the figure bellow +\begin{figure}[H] + \centering + \includegraphics[width=\textwidth]{"./plots/hosvd-uniform-error.png"} + \caption{Tucker approximation error on the $k-th$ step of a quasirandom + Tensor} +\end{figure} +\subsection{Tucker approximation of function-related tensors \label{sec: +repeat}} +Consider two multivariable functions $f(x_1, \dots, x_d)$ and $g(x_1, \dots, +x_d)$, defined as \begin{align} - \phi(V_1, \dots, V_d) = \left\|\Psi_r(V_1,\dots,V_d) - \Psi_R(U_1, - \dots,U_d) \right\|_F . + f(x_1, \dots, x_d) &= \left(1+\sum_{k=1}^d \frac{x^2_k}{8^{k-1}} + \right)^{-1}\\ + g(x_1, \dots, x_d) &= \sqrt{\sum_{k=1}^d \frac{x_k^2}{8^{k-1}}} \cdot + \left(1+\frac{1}{2}\cos\left(\sum_{k=1}^d \frac{4\pi x_k}{4^{k-1}}\right) \right) \end{align} -For $\Psi_r$ and $\Psi_R$ denote the CPD multilinear representation maps -transforming the CP decomposition into the tensor +for $x_1, \dots, x_d \in [-1, 1]$. Additionally we define a grid of points \begin{align} - U &= \Psi_R(U_1, \dots, U_d)\\ - V &= \Psi_r(V_1, \dots, V_d)\\ - \end{align} -for $k = 1, \dots d, d-1, \dots 2$ sequentially by updating $V_k$ at each -step of the optimization. \textbf{This is one iteration step}. - -For $\Psi$ we will use the implementation constructed in the last exercise, -which consists of applying the Kronecker product of the rows and then -summing over them. Once we define the following matrices we may rewrite the -optimality condition in a linear system which is solvable since it is a least -square problem. We define matrices $\mathcal{V}_k \in \mathbb{R}^{n_1\cdots n_d -\times n_k\cdot r}$ and $\mathcal{U}_k \in \mathbb{R}^{n_1\cdots n_d \times -n_k\cdot R}$ for $k \in \{1, \dots,d\}$ by the following + t_i = 2 \frac{i-1}{n-1} -1 , +\end{align} +for $i = 1, \dots, n$. With this we can construct a $d$ dimensional tensor of +size $n \times \cdots \times n$ by \begin{align} - \mathcal{U}_k &= \prod_{\substack{l=1 \\ l!=k}}^{d} U_l \otimes - \mathbbm{1}_{\mathbb{R}^{n_k\times n_k}},\\ - \mathcal{V}_k &= \prod_{\substack{l=1 \\ l!=k}}^{d} V_l \otimes - \mathbbm{1}_{\mathbb{R}^{n_k\times n_k}}, + b_{i_1, \cdots, i_d} = f(t_{i_1}, \dots, t_{i_d}),\\ + a_{i_1, \cdots, i_d} = g(t_{i_1}, \dots, t_{i_d}), \end{align} -note that $\prod$ represents the Hadamard product (elementwise multiplication) and -$\otimes$ the Kronecker product. The product of these two new defined -matrices is then simply +for all $i_1, \cdots, i_d \in \{1, \cdots, n\}$. + +For $C = A$ and $C =B$. +For every $k \in \{1, \dots , d\}$, we compute the singular values of the $k-th$ +Tucker unfolding matrix of C. +\begin{figure}[H] + \centering + \includegraphics[width=0.49\textwidth]{./plots/singular-dec-A.png} + \includegraphics[width=0.49\textwidth]{./plots/singular-dec-B.png} + \caption{Decrease of singular values of the $k-th$ Tucker unfolding of A, + B produced by its SVD} +\end{figure} + +For the accuracy Threshold $\eps_j = 10^{-j}$ with $j \in \{2, 4, \dots, +12\}$, for each $j$ and every $k$ we find the smallest $r_{jk}$ such that the +$k-th$ unfolding matrix can be approximated with ranks $r_{jk}$, where the +approximations relative error does not exceed $\eps_j$. Additionally we +compute the $\eps_{jk}$ error in the $k-th$ unfolding, which +should by the error analysis be bounded by $\eps_{jk} \leq \eps_{j}$ for all +$j$. +\begin{figure}[H] + \centering + \includegraphics[width=0.49\textwidth]{./plots/hosvd-error-A.png} + \includegraphics[width=0.49\textwidth]{./plots/hosvd-error-B.png} + \caption{Dependence of the approximated ranks $r_{jk}$ on $j = + \log_{10}\eps^{-j}$} +\end{figure} + +For every $j$ we use our implementation of the HOSVD algorithm to compute the +Tucker approximation of $C$ for the ranks $r_{jk}$ for $k = 1, \cdots ,d$ and +the number of total entries $N_j$ produced by the output decomposition. In +the code \cite{code} it is checked during run-time that the error produced by +the HOSVD algorithm does not exceed $\|\eps_{jk}\|_F$ and agrees wit the +error analysis. + +For $j = 12$ we plot the ratio of the singular values produced by the HOSVD +algorithm and during the SVD approximation of the $k-th$ Tucker unfolding +matrix +\begin{figure}[H] + \centering + \includegraphics[width=0.49\textwidth]{./plots/hosvd-sigmaratio-a.png} + \includegraphics[width=0.49\textwidth]{./plots/hosvd-sigmaratio-b.png} + \caption{Ratio of Singular values produced by the HOSVD and the $k-th$ + Tucker unfolding approximation of $C$.} +\end{figure} + +\begin{figure}[H] + \centering + \includegraphics[width=0.49\textwidth]{./plots/hosvd-Nj-a.png} + \includegraphics[width=0.49\textwidth]{./plots/hosvd-Nj-b.png} + \caption{Number of parameters $N_j$ produced by the HOSVD for $C$} +\end{figure} + + +\subsection{TT-SVD for MPS-TT} +The TT-MPS (Tensor Train or Matrix Product States decomposition) for a given +tensor $A \in \mathbb{R}^{n_1 \times \cdots \times n_d}$ with ranks $r_1, +\dots, r_{d-1} \in \mathbb{N}$ is the following \begin{align} - \mathcal{V}_k^*\mathcal{V}_k &= \prod_{\substack{l=1 \\ l!=k}}^{d} - (V_l^*V_l) \otimes \mathbbm{1}_{\mathbb{R}^{n_k\times n_k}} - \label{eq: VV} \\ - \mathcal{V}_k^*\mathcal{U}_k &= \prod_{\substack{l=1 \\ l!=k}}^{d} - (V_l^*U_l) \otimes \mathbbm{1}_{\mathbb{R}^{n_k\times n_k}}.\label{eq: - VU} + A_{i_1,\dots, i_d} = + \sum_{\alpha_1=1}^{r_1}\cdots\sum_{\alpha_{d-1}=1}^{r_{d-1}}U_1(\alpha_0,i_1, + \alpha_1) \cdots U_d(\alpha_{d-1}, i_d, \alpha_d) \end{align} -Do note that $\mathcal{V}_k^*\mathcal{V}_k \in \mathbb{R}^{r\cdot n_k \times -r\cdot n_k}$ and $\mathcal{V}_k^*\mathcal{U}_k \in \mathbb{R}^{r\cdot n_k -\times R\cdot n_k}$. For convenience let us define $F_k := V_k^* U_k$ and -$G_k := V_k^* V_k$, which will become useful when talking about storing the -computation. The we can solve for $V_k$ with the following least square -problem +for $\alpha_0 = \alpha_d = 1$, $i_k \in \{1, \dots, n_k\}$. The +decomposition factors are given by \begin{align} - (\mathcal{V}_k^*\mathcal{V}_k)\hat{v}_k = (\mathcal{V}_k^*\mathcal{U}_k)u_k, + V_k(i_k) &\in \mathbb{R}^{r_{k-1}\cdot n_k \times r_k}, \\ + (V_k(i_k))_{\alpha_{k-1}, \alpha_k} &= U_k (\alpha_{k-1}, i_k, \alpha_k), \end{align} -where $\hat{v}_k = \text{vec}(\hat{V}_k)$ and $u_k = \text{vec}(U_k)$. - -To make the computation cost linear with respect to $d$, we can compute -$\mathcal{V}_k^*\mathcal{V}_k$ and $\mathcal{V}_k^*\mathcal{U}_k$ for -each $k$ and update them in the iteration in $k$ with the new $\hat{V}_k$, by -computing the product in equations \ref{eq: VV} and \ref{eq: VU} again. -Additionally we compute the error $\phi$ and $\|\nabla \frac{1}{2} -\phi^2\|_2$ after each iteration (after going through -$k=2,\dots,d,d-1,\dots,1$). The implementation is in \cite{code}. - -\subsection{CP-ALS for the matrix multiplication tensor} -In this section we will use the matrix multiplication tensor to play around -the CP-ALS algorithm we implemented in the last section. We consider $n = 2$ -and $r = 7$, $n = 3$ and $r = 23$, $n = 4$ and $r = 49$ for the -multiplication tensor and its CP decomposition. The implementation of them we -have done in the last exercise together with their rank $R = n^3$ CP -decomposition. We test the our implementation of the CP-ALS algorithm of -these three multiplication tensor for three random initial guesses $V_1, V_2, -V_3$, where each matrix has unitary columns (i.e. of norm one). - -Our procedure will be that we do seven different guesses if they are good -enough, i.e. if they do not produce a \textit{SingularValueError} after 10 -iterations we go for 10000 iterations and plot both $\phi$ and $\|\nabla -\frac{1}{2} \phi\|_2$ with respect to the number of iterations for each -multiplication tensor. We get the following curves +where $i_k$ is called the mode index. + +The TT-SVD algorithm for $A$ as above is the following +\begin{algorithm}[H] + \caption{TT-SVD algorithm}\label{alg: ttsvd} +\begin{algorithmic} + \State $\hat{S}_0 \gets A$ + \State $A_0 \gets A$ + \State $r_0 \gets 1$ + \State $r_d \gets 1$ + \For{$k = 1, \dots, d-1$} + \State $(B_k)_{\alpha_{k-1}\cdot i_k, i_{k+1} \cdots i_d} \gets + (\hat{S}_{k-1})_{\alpha_{k-1}, i_k, i_{k+1}, \dots, i_d}$ + \Comment{reshape} + + \State $\hat{B}_k \gets \hat{U}_k \hat{\Sigma}_k \hat{V}_k^*$ + \Comment{rank $r_k$ T-SVD for $B_k$} + + \State $(\hat{C}_k)_{\alpha_{k-1},i_k,\alpha_k} \gets + (\hat{U}_k)_{\alpha_{k-1}i_k, \alpha_k}$ \Comment{reshape} + + \State $(\hat{S}_k)_{\alpha_k, i_k, \dots, i_d} \gets + (\hat{\Sigma}_k\hat{V}_k)_{\alpha_k, i_{k+1} \dots i_d}$ + \Comment{reshape} + + \State \text{save} $\frac{\| A - \hat{A}_k\|_F}{\|A\|_F}$ + \State \text{save} $\hat{C}_k$ + \State \text{save} $\hat{\Sigma}_k$ + \EndFor +\end{algorithmic} +\end{algorithm} +\subsection{Testing the TT-SVD} +For the case $d=4$ we construct a quasirandom Tucker decomposition by drawing +the entries for $C_k \in \mathbb{R}^{r_{k-1} \cdot n_k \times r_k}$ uniformly +on $[-1, 1]$. The output of the errors in the $k-th$ steps is in the figure +bellow \begin{figure}[H] - \centering \includegraphics[width=0.33\textwidth]{./../src/plots/err_2.png} - \includegraphics[width=0.33\textwidth]{./../src/plots/err_3.png} - \includegraphics[width=0.32\textwidth]{./../src/plots/err_4.png} - \caption{From left to right, rank $r\in\{7, 23, 49\}$ CP-ALS error for the - $n\in \{2, 3, 4\}$ multiplication tensor at 10000 iteration steps} + \centering + \includegraphics[width=\textwidth]{"./plots/ttsvd-uniform-error.png"} + \caption{TT-MPS error on the $k-th$ step of a quasirandom Tensor in the + TT-SVD algorithm} \end{figure} +\subsection{TT-MPS of function-related tensors} +In this section we repeat everything we did in section \ref{sec: repeat}, +replacing the HOSVD algorithm for the Tucker approximation with the TT-SVD +algorithm for the TT-MPS approximation +\begin{figure}[H] + \centering + \includegraphics[width=0.49\textwidth]{./plots/ttsvd-sigmaratio-a.png} + \includegraphics[width=0.49\textwidth]{./plots/ttsvd-sigmaratio-b.png} + \caption{Ratio of Singular values produced by the TT-SVD and the $k-th$ + Tucker unfolding approximation of $C$.} +\end{figure} + +\begin{figure}[H] + \centering + \includegraphics[width=0.49\textwidth]{./plots/ttsvd-Nj-a.png} + \includegraphics[width=0.49\textwidth]{./plots/ttsvd-Nj-b.png} + \caption{Number of parameters $N_j$ produced by the TT-SVD for $C$} +\end{figure} +\nocite{code} \printbibliography \end{document} diff --git a/sesh4/tex/plots/hosvd-Nj-a.png b/sesh4/tex/plots/hosvd-Nj-a.png Binary files differ. diff --git a/sesh4/tex/plots/hosvd-Nj-b.png b/sesh4/tex/plots/hosvd-Nj-b.png Binary files differ. diff --git a/sesh4/tex/plots/hosvd-error-A.png b/sesh4/tex/plots/hosvd-error-A.png Binary files differ. diff --git a/sesh4/tex/plots/hosvd-error-B.png b/sesh4/tex/plots/hosvd-error-B.png Binary files differ. diff --git a/sesh4/tex/plots/hosvd-sigmaratio-a.png b/sesh4/tex/plots/hosvd-sigmaratio-a.png Binary files differ. diff --git a/sesh4/tex/plots/hosvd-sigmaratio-b.png b/sesh4/tex/plots/hosvd-sigmaratio-b.png Binary files differ. diff --git a/sesh4/tex/plots/hosvd-uniform-error.png b/sesh4/tex/plots/hosvd-uniform-error.png Binary files differ. diff --git a/sesh4/tex/plots/singular-dec-A.png b/sesh4/tex/plots/singular-dec-A.png Binary files differ. diff --git a/sesh4/tex/plots/singular-dec-B.png b/sesh4/tex/plots/singular-dec-B.png Binary files differ. diff --git a/sesh4/tex/plots/ttsvd-Nj-a.png b/sesh4/tex/plots/ttsvd-Nj-a.png Binary files differ. diff --git a/sesh4/tex/plots/ttsvd-Nj-b.png b/sesh4/tex/plots/ttsvd-Nj-b.png Binary files differ. diff --git a/sesh4/tex/plots/ttsvd-sigmaratio-a.png b/sesh4/tex/plots/ttsvd-sigmaratio-a.png Binary files differ. diff --git a/sesh4/tex/plots/ttsvd-sigmaratio-b.png b/sesh4/tex/plots/ttsvd-sigmaratio-b.png Binary files differ. diff --git a/sesh4/tex/plots/ttsvd-uniform-error.png b/sesh4/tex/plots/ttsvd-uniform-error.png Binary files differ.
